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Suppose we observe data $y_t$ and $X_t$ from $t=1,...,T$ and want to estimate a dynamic linear model of the form

$y_t = X_t\beta_t + \epsilon_t$

$\beta_t = \beta_{t-1} + \omega_t$

where $\epsilon_t$ and $\omega_t$ are normally distributed with zero mean and known variances. Suppose that the initial value $\beta_0$ follows a normal distribution with known moments s.t. $\beta_0 \sim N(a_0,P_0)$.

The state sequence $\beta_1,...,\beta_T$ can easily be estimated using the Kalman filter and Kalman smoother that are for instance outlined in the book of Durbin & Koopman. They provide the necessary forward filtering and backward smoothing recursions for the time periods $1,...,T$.

Usually, one would initialize the forward filtering procedure by setting the filtered mean to $a_0$ and the filtered variance to $P_0$ in period $t=0$. This initialization then allows us to forward filter from $t=1$ up to $t=T$.

Then, the Kalman smoother updates the filtered state means and variances using backward recursions, going from $t=T$ to $t=1$. All of this works fine and the estimates are looking good. However, I would like to also update $a_0$ and $P_0$ in the backward smoothing process. How can we do that? If I follow the usual backward recursions, it appears that I would need data in $t=0$ to do so.

Edit:

I think the question is equivalent to:

Are the filtered estimates at $t=T$ equvialent to the smoothed estimates at $t=T$ and do we therefore have to do the backward recursions from $t-1$ to $0$ after initializing the smoothed estimates at $T$ using the filtered estimates at $T$?

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